Let $$\overrightarrow \alpha $$ = $$\widehat i + \widehat j + \widehat k$$, $$\overrightarrow \beta $$ = $$\widehat i - \widehat j - \widehat k$$ and $${\overrightarrow \gamma }$$ = $$ - \widehat i - \widehat j - \widehat k$$ be three vectors. A vector $$\overrightarrow \delta $$, in the plane of $$\overrightarrow \alpha $$ and $$\overrightarrow \beta $$, whose projection on $${\overrightarrow \gamma }$$ is $${1 \over {\sqrt 3 }}$$, is given by

Let $$\overrightarrow \alpha $$, $${\overrightarrow \beta }$$, $${\overrightarrow \gamma }$$ be the three unit vectors such that $$\overrightarrow \alpha .\overrightarrow \beta = \overrightarrow \alpha .\overrightarrow \gamma = 0$$ and the angle between $$\overrightarrow \beta $$ and $$\overrightarrow \gamma $$ is 30$$^\circ$$. Then $$\overrightarrow \alpha $$ is

For any vector x, where $$\widehat i$$, $$\widehat j$$, $$\widehat k$$ have their usual meanings the value of $${(x \times \widehat i)^2} + {(x \times \widehat j)^2} + {(x \times \widehat k)^2}$$ where $$\widehat i$$, $$\widehat j$$, $$\widehat k$$ have their usual meanings, is equal to

If the sum of two unit vectors is a unit vector, then the magnitude of their difference is